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We settle an open problem of several years standing by showing that the least-squares mean for positive definite matrices is monotone for the usual (Loewner) order.
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Y. Lim, Applications of geometric means on symmetric cones, Math. Ann. 319
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2006
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X. Pennec, P. Fillard, and N. Ayache, A Riemannian framework for tensor computing, International Journal of Computer Vision, 66
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E. Ahn, S. Kim, and Y. Lim, An extended Lie-Trotter formula and its applications, Linear Algebra Appl. 427
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R. Bhatia, Positive definite matrices
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P. D. Fletcher and S. Joshi, Riemannian geometry for the statistical analysis of difusion tensor data, Signal Processing 87
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2003
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T. Ando, C.K. Li and R. Mathias, Geometric means, Linear Algebra Appl. 385
2004
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M. Moakher, A differential geometric approach to the geometric mean of symmetric positive-definite matrices, SIAM J. Matrix Anal. Appl. 26
2005
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R. Bhatia and J. Holbrook, Noncommutative geometric means, Math. Intelligencer 28
2006
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J. D. Lawson, H. Lee and Y. Lim, Weighted barycenters on metric spaces, preprint
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T. Yamazaki, On properties of geometric mean of n n -operators via Riemannian metric, preprint
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M. Zerai and M. Moakher, The Riemannian geometry of the space of positive-deÞnite matrices and its application to the regularization of di?usion tensor MRI data, J. Math. Imaging Vision, submitted
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F. Barbaresco, Innovative tools for radar signal processing based on Cartan’s geometry of symmetric positive definite matrices and information geometry, in Proceedings of the IEEE International Radar Conference, Rome, Italy, 2008
2008
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J. D. Lawson and Y. Lim, A general framework for extending means to higher orders, Colloq. Math. 113
2008
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X. Zhan, Open problems in matrix theory, in: Proceedings of the 4 4 th International Congress of Chinese Mathematicians
2008
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D. Bini, B. Meini and F. Poloni, An effective matrix geometric mean satisfying the Ando-Li-Mathias properties, Math. Comp. 79
2010
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